Section: New Results

Dynamic Neural Fields

Participants : Lucian Aleçu, Frédéric Alexandre, Yann Boniface, Laurent Bougrain, Mauricio Cerda, Georgios Detorakis, Hervé Frezza-Buet, Bernard Girau, Axel Hutt, Mathieu Lefort, Jean-Charles Quinton, Nicolas Rougier, Wahiba Taouali, Thierry Viéville, Thomas Voegtlin.

The work reported this year represents both extensions of previous works and new results linked to the notion of neural population, considered at (i) a formal level (theorical studies of neural fields), (ii) a numerical level (study of functioning and learning rules) and (iii) a more embodied one (implementations of specific functions).

Formal Level

• study of the differences between synchronous and asynchronous (without a central clock) evaluation: The hallmark of most artificial neural networks is their supposed intrinsic parallelism where each unit is evaluated concurrently to other units in a distributed way. However, if one gives a closer look under the hood, one can soon realize that such a parallelism is an illusion since most implementations use what is referred to as synchronous evaluation, or using a central clock. Here we propose to consider different evaluation methods (namely asynchronous and event based evaluation methods) and study their properties in some restricted but illustrative cases. This work is also in preparation for publication.

• taking into account transmission speed between units in a neural field: Neurons in populations are connected to each other by axonal branches sending electric pulses. The pulse propagation with finite speed delays the neuron interactions. The developed numerical algorithm illustrates how to simulate neural fields in two spatial dimensions involving finite axonal transmission speeds. The algorithm is derived analytically shows how to implement a Fast Fourier Transform in the computation scheme.

• study of the bridge between an ensemble of spiking neurons and the population firing rate to extend neural fields by shunting inhibition effects. Shunting inhibition is an important effect in real neural systems, e.g. in the context of general anaesthesia. We re-derive the population firing rate well-known in neural fields from the single neuron firing statistics. This derivation assumes McCulloch-Pitts neurons with a trivial f-I curve. Then we exchange the McCulloch-Pitts neurons by more realistic type I-neurons with a non-trivial f-I curve and gain a different, more realistic population firing rate. This formulation allows to consider some shunting inhibition effects [44] .

Numerical Level

Numerical studies of DNF and related mechanisms

At the numerical level, specific developments were carried out to assess our software platform, to master functioning rules and to study the performances of new learning rules:

• The problem of adjusting the parameters of a mesoscopic event and valued neural field with delayed connections is addressed here at the programmatic level. An effective computational framework, with the implementation of a general algorithm is developed allowing us to effectively design non-trivial input/output transformations of events and values, using a class of biologically plausible distributed functional models. This work is in preparation for publication.

• In order to clarify the notion of distributed computing, general concepts and definitions in the framework of artificial neural networks have been reviewed, within the scope of dynamic field theory, proposing an unequivocal definition of asynchronous computation. An innovative way to perform such asynchronous computation has been proposed, following theoretical developments in process formalization. Several consequences on both the trajectories and the stability of the whole system have been drawn, including a few practically usable methods and quantitative bounds that can guarantee most of the mesoscopic properties of the system [15] .

• Novel numerically efficient algorithm to compute spatio-temporal activity in two-dimensional neural fields involving finite transmission speed.

• Study of the possibility to obtain properties of self-organization with dynamic neural fields and proposition of a new learning rule for self-organization [1] , [5] .

• We designed a variation of the self-organising map algorithm [14] where the original time-dependent (learning rate and neighbourhood) learning function has been replaced by a time-invariant one. This allows for on-line and continuous learning on both static and dynamic data distributions. One of the property of the newly proposed algorithm is that it does not fit the magnification law and the achieved vector density is not directly proportional to the density of the distribution as found in most vector quantisation algorithms. From a biological point of view, this algorithm sheds light on cortical plasticity seen as a dynamic and tight coupling between the environment and the model.

• Adaptation of the BCM rule to multi-modality by adapting the dynamics of the threshold by the use of a feed-back signal generated by a neural field map [22] , [39] , [45]

• Following [25] , we are now studying a computational model of the primary somato-sensory cortex based on the neural field theory where cortical representations develop through the modification of thalamocortical synapses (from thalamus to layer 4), while cortico-cortical synapses (layer 2/3 and 4) provide a distributed competitive mechanism between cortical pyramidal neurons of layer 2/3. Preliminary results explains both the initial development and the self-organization of cortical representations in the primary sensory cortex as well as the dynamic reorganization following a lesion or a sensory deprivation. In this context, the so-called critical period during childhood would correspond to the development and learning of the intra-cortical competitive mechanism that is critical for cortex plasticity.

Gaussian mixture based approximation of neural maps

We have studied the advantages of our new implementation of the Continuous Neural Field Theory (CNFT) using a Gaussian mixture based model of the neural field activity, when using high dimensional inputs [40] . It exploits the rapid convergence of the activity to a reduced set of localized bubbles when competition occurs. These bubbles of activity can be accurately approximated by Gaussian distributions, that are directly computed in any n-dimensional space, instead of projecting high dimensional inputs onto 2D maps (which generally leads to topological distortions). This implementation is thus used to evaluate the possibilities of sensorimotor or multimodal associations without prior self-organization on 2D cortical maps, and could be directly interfaced with high dimensional artificial systems.

Embodied Level

Motion detection

We develop bio-inspired neural architectures to extract and segment the direction and speed components of the optical flow from sequences of images. Following this line, we have recently built additional models to code and distinguish different visual sequences. The structure of these models takes inspiration from the course of visual movement processing in the human brain, such as in area MT (middle temporal) that detects patterns of movement, or area FBA where neurons have been found to be sensitive to single spatio-temporal patterns. This work has been recently extended to complex movements: to fight, to wave, to clap, using real-world video databases [2] , as well as using speech-driven visual animations of faces [28] .

Modeling the superior colliculus by mean of a neural field.

In the context of the ANR MAPS project (cf. §  8.2 ), we have been studying the superior colliculus in tight collaboration with Laurent Goffart from the Institut de Neurosciences Cognitives de la Méditerranée. Considering the cortical magnification induced by the non homogeneous distribution of retina rods and cones on the retina surface, we modeled the superior colliculus using a dynamic neural field that may explain the stereotyped nature of colliculus activity. This year, we have extended this approach to wider contexts:

• Using Neural Fields to model the Superior Colliculus in a task of saccade generation

• Arrangement of several neural fields to model several cortical areas engaged in visual attention

Modeling of neural activity during anaesthesia.

Anaesthesia plays an important role in medical surgery though its neural mechanism is still poorly understood. Besides several different molecular and behavioral phenomena, the administration of anaesthetic agents affects the power spectrum of electro-encephalographic activity (EEG) in a characteristic way. The theoretical study aims to model the power spectrum changes in EEG subject to the concentration of the specific anaesthetic agent propofol. The work developed a neural model [38] involving two neuron types and synapse types while taking into account the synaptic effect of propofol. The mathematical derivation of the power spectrum allows for the investigation of suitable physiological parameters which reproduce the experimental effect of propofol. Several mathematical conditions on physiological parameters have been derived and the EEG-power spectrum during the administration of different concentration levels of propofol has been modeled successfully.